Theorem ipasslem1 27070

Description: Lemma for ipassi 27080. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip1i.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHilOLD
ipasslem1.b	⊢ 𝐵 ∈ 𝑋

Assertion

Ref	Expression
ipasslem1	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Proof of Theorem ipasslem1

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 11179	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
2		ax-1cn 9873	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
3		ip1i.9	. . . . . . . . . . . . . 14 ⊢ 𝑈 ∈ CPreHilOLD
4	3	phnvi 27055	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = (BaseSet‘𝑈)
6		ip1i.2	. . . . . . . . . . . . . 14 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
7		ip1i.4	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
8	5, 6, 7	nvdir 26870	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
9	4, 8	mpan 702	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
10	2, 9	mp3an2 1404	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
11	1, 10	sylan 487	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)))
12	5, 7	nvsid 26866	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
13	4, 12	mpan 702	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (1𝑆𝐴) = 𝐴)
14	13	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
15	14	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘𝑆𝐴)𝐺(1𝑆𝐴)) = ((𝑘𝑆𝐴)𝐺𝐴))
16	11, 15	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1)𝑆𝐴) = ((𝑘𝑆𝐴)𝐺𝐴))
17	16	oveq1d 6564	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵))
18		ipasslem1.b	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ 𝑋
19		ip1i.7	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
20	5, 19	dipcl 26951	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
21	4, 18, 20	mp3an13 1407	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (𝐴𝑃𝐵) ∈ ℂ)
22	21	mulid2d 9937	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → (1 · (𝐴𝑃𝐵)) = (𝐴𝑃𝐵))
23	22	adantl 481	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (1 · (𝐴𝑃𝐵)) = (𝐴𝑃𝐵))
24	23	oveq2d 6565	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
25	5, 7	nvscl 26865	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑘𝑆𝐴) ∈ 𝑋)
26	4, 25	mp3an1 1403	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑘𝑆𝐴) ∈ 𝑋)
27	1, 26	sylan 487	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (𝑘𝑆𝐴) ∈ 𝑋)
28	5, 6, 7, 19, 3	ipdiri 27069	. . . . . . . . . . 11 ⊢ (((𝑘𝑆𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
29	18, 28	mp3an3 1405	. . . . . . . . . 10 ⊢ (((𝑘𝑆𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
30	27, 29	sylancom 698	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (𝐴𝑃𝐵)))
31	24, 30	eqtr4d 2647	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))) = (((𝑘𝑆𝐴)𝐺𝐴)𝑃𝐵))
32	17, 31	eqtr4d 2647	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))))
33		oveq1 6556	. . . . . . 7 ⊢ (((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵)) → (((𝑘𝑆𝐴)𝑃𝐵) + (1 · (𝐴𝑃𝐵))) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
34	32, 33	sylan9eq 2664	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
35		adddir 9910	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
36	2, 35	mp3an2 1404	. . . . . . . 8 ⊢ ((𝑘 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
37	1, 21, 36	syl2an 493	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
38	37	adantr 480	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → ((𝑘 + 1) · (𝐴𝑃𝐵)) = ((𝑘 · (𝐴𝑃𝐵)) + (1 · (𝐴𝑃𝐵))))
39	34, 38	eqtr4d 2647	. . . . 5 ⊢ (((𝑘 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))
40	39	exp31 628	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝐴 ∈ 𝑋 → (((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵)) → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))))
41	40	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝐴 ∈ 𝑋 → ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))) → (𝐴 ∈ 𝑋 → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))))
42		eqid 2610	. . . . . 6 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
43	5, 42, 19	dip0l 26957	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0)
44	4, 18, 43	mp2an 704	. . . 4 ⊢ ((0vec‘𝑈)𝑃𝐵) = 0
45	5, 7, 42	nv0 26876	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (0vec‘𝑈))
46	4, 45	mpan 702	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (0𝑆𝐴) = (0vec‘𝑈))
47	46	oveq1d 6564	. . . 4 ⊢ (𝐴 ∈ 𝑋 → ((0𝑆𝐴)𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵))
48	21	mul02d 10113	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (0 · (𝐴𝑃𝐵)) = 0)
49	44, 47, 48	3eqtr4a 2670	. . 3 ⊢ (𝐴 ∈ 𝑋 → ((0𝑆𝐴)𝑃𝐵) = (0 · (𝐴𝑃𝐵)))
50		oveq1 6556	. . . . . 6 ⊢ (𝑗 = 0 → (𝑗𝑆𝐴) = (0𝑆𝐴))
51	50	oveq1d 6564	. . . . 5 ⊢ (𝑗 = 0 → ((𝑗𝑆𝐴)𝑃𝐵) = ((0𝑆𝐴)𝑃𝐵))
52		oveq1 6556	. . . . 5 ⊢ (𝑗 = 0 → (𝑗 · (𝐴𝑃𝐵)) = (0 · (𝐴𝑃𝐵)))
53	51, 52	eqeq12d 2625	. . . 4 ⊢ (𝑗 = 0 → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ ((0𝑆𝐴)𝑃𝐵) = (0 · (𝐴𝑃𝐵))))
54	53	imbi2d 329	. . 3 ⊢ (𝑗 = 0 → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → ((0𝑆𝐴)𝑃𝐵) = (0 · (𝐴𝑃𝐵)))))
55		oveq1 6556	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗𝑆𝐴) = (𝑘𝑆𝐴))
56	55	oveq1d 6564	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝑗𝑆𝐴)𝑃𝐵) = ((𝑘𝑆𝐴)𝑃𝐵))
57		oveq1 6556	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑗 · (𝐴𝑃𝐵)) = (𝑘 · (𝐴𝑃𝐵)))
58	56, 57	eqeq12d 2625	. . . 4 ⊢ (𝑗 = 𝑘 → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵))))
59	58	imbi2d 329	. . 3 ⊢ (𝑗 = 𝑘 → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → ((𝑘𝑆𝐴)𝑃𝐵) = (𝑘 · (𝐴𝑃𝐵)))))
60		oveq1 6556	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝑗𝑆𝐴) = ((𝑘 + 1)𝑆𝐴))
61	60	oveq1d 6564	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((𝑗𝑆𝐴)𝑃𝐵) = (((𝑘 + 1)𝑆𝐴)𝑃𝐵))
62		oveq1 6556	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 · (𝐴𝑃𝐵)) = ((𝑘 + 1) · (𝐴𝑃𝐵)))
63	61, 62	eqeq12d 2625	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵))))
64	63	imbi2d 329	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → (((𝑘 + 1)𝑆𝐴)𝑃𝐵) = ((𝑘 + 1) · (𝐴𝑃𝐵)))))
65		oveq1 6556	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑗𝑆𝐴) = (𝑁𝑆𝐴))
66	65	oveq1d 6564	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑗𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
67		oveq1 6556	. . . . 5 ⊢ (𝑗 = 𝑁 → (𝑗 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
68	66, 67	eqeq12d 2625	. . . 4 ⊢ (𝑗 = 𝑁 → (((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵)) ↔ ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))))
69	68	imbi2d 329	. . 3 ⊢ (𝑗 = 𝑁 → ((𝐴 ∈ 𝑋 → ((𝑗𝑆𝐴)𝑃𝐵) = (𝑗 · (𝐴𝑃𝐵))) ↔ (𝐴 ∈ 𝑋 → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))))
70	41, 49, 54, 59, 64, 69	nn0indALT 11349	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ 𝑋 → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))))
71	70	imp 444	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℕ₀cn0 11169  NrmCVeccnv 26823   +_𝑣 cpv 26824  BaseSetcba 26825   ·_𝑠OLD cns 26826  0_veccn0v 26827  ·_𝑖OLDcdip 26939  CPreHil_OLDccphlo 27051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-dip 26940  df-ph 27052

This theorem is referenced by:  ipasslem2  27071  ipasslem3  27072  ipasslem4  27073